2. CONTENTS
● The Basics of Counting
● The Pigeonhole Principle
● Permutations and Combinations
● Binomial Coefficients and Identities
CONCLUSION
●
3. Basic Counting Principles: The
Product Rule
The Product Rule: A procedure can be
broken down into a sequence of two
tasks. There are n1 ways to do the first
task and n2 ways to do the second task.
Then there are n1·n2 ways to do the
procedure.
Example: How many bit strings of length seven
are there?
Solution: Since each of the seven bits is either
a 0 or a 1, the answer is 27 = 128.
4. Basic Counting Principles:
The Sum Rule
The Sum Rule: If a task can be done either in one of n1 ways or in
one of n2 ways to do the second task, where none of the set of
n1 ways is the same as any of the n2 ways, then there are n1 +
n2 ways to do the task.
Example: The mathematics department must choose either a
student or a faculty member as a representative for a university
committee. How many choices are there for this representative
if there are 37 members of the mathematics faculty and 83
mathematics majors and no one is both a faculty member and a
student.
Solution: By the sum rule it follows that there are 37 + 83 = 120
possible ways to pick a representative.
5. Basic Counting Principles:
Subtraction Rule
Basic Counting Principles:
Division Rule
Division Rule: There are n/d ways to do a task if it can be done using a
procedure that can be carried out in n ways, and for every way w,
exactly d of the n ways correspond to way w.
● Restated in terms of sets: If the finite set A is the union of n
pairwise disjoint subsets each with d elements, then n = |A|/d.
● In terms of functions: If f is a function from A to B, where both are
finite sets, and for every value y ∈ B there are exactly d values x ∈
A such that f(x) = y, then |B| = |A|/ d.
Subtraction Rule: If a task can be done either
in one of n1 ways or in one of n2 ways, then
the total number of ways to do the task is n1 +
n2 minus the number of ways to do the task
that are common to the two different ways.
6. The Pigeonhole Principle
If a flock of 20 pigeons roosts in a set of 19
pigeonholes, one of the pigeonholes must
have more than 1 pigeon.
Pigeonhole Principle: If k is a positive integer
and k + 1 objects are placed into k boxes,
then at least one box contains two or more
objects.
Proof: We use a proof by contraposition.
Suppose none of the k boxes has more than
one object. Then the total number of
objects would be at most k. This contradicts
the statement that we have k + 1 objects.
7. Corollary 1: A function f from a set with k + 1 elements to a set
with k elements is not one-to-one.
Proof: Use the pigeonhole principle.
● Create a box for each element y in the codomain of f .
● Put in the box for y all of the elements x from the domain such that
f(x) = y.
● Because there are k + 1 elements and only k boxes, at least one box
has two or more elements.
Hence, f can’t be one-to-one.
The Generalized Pigeonhole Principle: If N objects are placed into k
boxes, then there is at least one box containing at least ⌈N/k⌉
objects.
Proof: We use a proof by contraposition. Suppose that none of
the boxes contains more than ⌈N/k⌉ − 1 objects. Then the total
number of objects is at most
where the inequality ⌈N/k⌉ < ⌈N/k⌉ + 1 has been used. This is a
contradiction because there are a total of n objects.
8. Permutations
Definition: A permutation of a set of distinct objects is an
ordered arrangement of these objects. An ordered
arrangement of r elements of a set is called an
r-permutation.
Example: Let S = {1,2,3}.
● The ordered arrangement 3,1,2 is a permutation of S.
● The ordered arrangement 3,2 is a 2-permutation of S.
Theorem 1: If n is a positive integer and r is an integer with
1 ≤ r ≤ n, then there are
P(n, r) = n(n − 1)(n − 2) ··· (n − r + 1) r-
permutations of a set with n distinct elements.
Proof: Use the product rule. The first element can be chosen in n
ways. The second in n − 1 ways, and so on until there are (n
− ( r − 1)) ways to choose the last element.
● Note that P(n,0) = 1, since there is only one way to order zero
elements.
Corollary 1: If n and r are integers with 1 ≤ r ≤ n, then
9. Combinations
Definition: An r-combination of elements of a set is an
unordered selection of r elements from the set. Thus, an
rcombination is simply a subset of the set with r elements.
Example: Let S be the set {a, b, c, d}. Then {a, c, d} is a
3combination from S. It is the same as {d, c, a} since the order
listed does not matter.
● C(4,2) = 6 because the 2-combinations of {a, b, c, d} are the six
subsets {a, b}, {a, c}, {a, d}, {b, c}, {b, d}, and {c, d}.
Example: A group of 30 people have been trained as astronauts to go on
the first mission to Mars. How many ways are there to select a crew of
six people to go on this mission?
Solution: By Theorem 2, the number of possible crews is
Example: How many ways are there to select five players from a 10-
member tennis team to make a trip to a match at another school.
Solution: By Theorem 2, the number of combinations is
10. BINOMIAL COEFFICIENT
Binomial Theorem: Let x and y be variables, and n a nonnegative
integer. Then:
Proof: We use combinatorial reasoning . The terms in the
expansion of (x + y)n are of the form xn−jyj for j = 0,1,2,…,n.
To form the term xn−jyj, it is necessary to choose n−j xs from the n
sums. Therefore, the coefficient of xn−jyj is which equals
Definition: A binomial expression is the sum of two terms, such as x
+ y. (More generally, these terms can be products of constants
and variables.)
● We can use counting principles to find the coefficients in the
expansion of (x + y)n where n is a positive integer.
● To illustrate this idea, we first look at the process of expanding (x
+ y)3.
11. CONCLUSION
The Fundamental Counting Principle is the quickest
and easiest way to calculate the total amount of
possible outcomes. It can be used in everyday life in
several aspects. From determining how many outfits
you can make with your clothes, to choosing a color
of your preference on a certain object. This principle
is commonly used in advertisements of Ice Cream
Shops that inform people what combinations they can
make with the flavors.
